A phenomenological model for analyzing reciprocating compressors

International Journal of Refrigeration 30 (2007) 1254e1265 www.elsevier.com/locate/ijrefrig A phenomenological model for analyzing reciprocating compressors E. Navarroa,*, E. Granrydb, J.F. Urchueguı́aa, J.M. Corberana a Universidad Politécnica de Valencia, Instituto de Ingenierı́a Energética, Investigación y Modelado de Sistemas Térmicos, Camino de Vera, 14, 46022 Valencia, Spain b Royal Institute of Technology, Department of Energy Technologies, Division of Applied Thermodynamics and Refrigeration, 100 44 Stockholm, Sweden Received 10 May 2006; received in revised form 25 January 2007; accepted 5 February 2007 Available online 24 February 2007 Abstract A new model for hermetic reciprocating compressors is presented. This model is able to predict compressor efficiency and volumetric efficiency in terms of a certain number of parameters (10) representing the main sources of losses inside the compressor. The model provides users with helpful information about the way in which the compressor is designed and working. A statistical fitting procedure based on the Monte Carlo method was developed for its adjustment. The model can predict compressor performance at most points with a maximum deviation of 3%. A possible gas condensation on cold spots inside the cylinder during the last part of the compression stroke was also evaluated. Ó 2007 Elsevier Ltd and IIR. All rights reserved. Keywords: Air conditioning; Refrigeration; Reciprocating compressor; Hermetic compressor; Modelling; Efficacy; Volumetric efficiency; Performance; Condensation; Gas Modèle d’analyse pour les compresseurs à piston fondé sur plusieurs phénomènes Mots clés : Conditionnement d’air ; Réfrigération ; Compresseur à piston ; Compresseur hermétique ; Modélisation ; Efficacité ; Rendement volumétrique ; Performance ; Condensation ; Gaz 1. Introduction * Corresponding author. Tel.: þ34 96 387 98 95; fax: þ34 96 387 95 29. E-mail address: [email protected] (E. Navarro). 0140-7007/$35.00 Ó 2007 Elsevier Ltd and IIR. All rights reserved. doi:10.1016/j.ijrefrig.2007.02.006 Reciprocating hermetic compressors have been known since the 19th century and, due to their simplicity and flexibility when working in a wide range of conditions, they are still used nowadays in refrigeration and air conditioning systems. E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 1255 Nomenclature (all the variables are expressed in SI units) AL Av cpi D dh E_ k _ mech EL Dhð1e8 Þ Dh(4e5) hfg hht hpc k Ki m_ in m_ leak m_ pc n nz Pi R Rp S Ti Leakage effective area Area for phase change inside the cylinder Specific heat at constant pressure in state i Cylinder diameter Effective hydraulic diameter Electric power consumption Energy lost in mechanical losses Enthalpy difference between states 1 and 8* Enthalpy difference between states 4 and 5 Heat of vaporization Heat transfer coefficient per area for the heat transfer between the hot and the cold gas Heat transfer coefficient per area for phase change inside the cylinder Thermal diffusivity Representative parameters for different model losses Mass flow rate Mass flow rate leaked Mass flow rate in phase change Compressor nominal speed Number of cylinders Pressure in state i Universal gas constant Pressure ratio Cylinder stroke Temperature in state i A complete empirical characterization procedure for this type of compressor is described in Ref. [1]. However, a certain theoretical effort based on analysis and modelling may be useful at this point to estimate how a given compressor is likely to work under different operating conditions or with different refrigerants, but also in more general terms, to assess the proper operation of the compressor. Two main categories of models are discussed in the literature:  Models whose aim is to explain in a detailed and accurate manner, the behavior of specific parts or processes (mechanics of the valves, vibration, heat transfer) inside the compressor. In the Proceedings of the International Compressor Engineering Conferences at Purdue [2] one can find numerous examples of models of this type including [3e6] and the like. Here the main objective is to assist the compressor optimization.  Models whose objective is to describe the compressor globally. In this category, three main basic approaches to the problem can be established: (1) Correlations from experimental data for some of the compressor significant variables such as COP and cooling capacity [1,7]. This is the methodology DTsh (UA)ht Vd Vs V_ s w Wref Zelvap Zmvap Compressor inlet gas superheat Overall heat transfer coefficient per area for the heat transfer between the hot and the cold gas Dead space Swept volume Compressor swept volume flow Refrigerant velocity Ideal isentropic work transferred to refrigerant by compressor Fraction of electric motor losses going to the suction gas Fraction of mechanical losses going to the suction gas Greek symbols hel Electric efficiency hk Compressor efficiency hs Volumetric efficiency hsth Theoretical volumetric efficiency m Dynamic viscosity ri Density in state i rin Density in compressor inlet t Fraction of time in which condensation can be produced x Drag factor for compressor inlet and outlet valves z Equivalent flow resistance for leakages most commonly used by compressor manufacturers, but it does not give any valuable physical information about the processes inside the compressor. The correlations obtained can only be used for the range of conditions in which they were obtained. (2) Other authors [8e11] have attempted to model the most important physical compressor processes using numerical methods to solve the differential equations implied in the conservation laws of these processes. Although these kinds of models may deliver considerable information about the way in which the compressor is working, they usually require numerous data available only to the manufacturer. These models aim to optimize compressor design. (3) The so-called semi-empirical models, for instance [12e14], seek to reproduce the main compressor performance variables like COP and cooling capacity using empirically adjusted, simple models retaining at least some portion of the physical background. Given their simplicity, these models do not need as much information as the detailed models described in approach (2). As a consequence, the information obtained is not as accurate, 1256 E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 and there are usually problems in the physical interpretation of certain results. This research describes a global analysis of a series of hermetic reciprocating compressors covering different strokes, piston numbers, sizes and refrigerants. The results of this research were divided into two papers. In this first paper, a new phenomenological model which aims to identify the most important phenomena occurring inside the compressor and the corresponding empirical coefficients required for their adjustment are described. In the second paper [15], the resulting adjustment of the model to experimental results from a vast experimental test campaign is discussed, and the empirical coefficients obtained are analyzed. Finally, a full discussion about the physical sense and interpretation of the obtained values, as well as the capabilities of the model to help in the analysis of the behavior and internal characteristics of the studied compressors is presented. The proposed compressor model aims to reproduce the compressor efficiency ðhk ¼ Wref =E_ k Þ and the volumetric efficiency ðhs ¼ m_ in =V_ s rin Þ as a function of a set of parameters which may be obtained by correlations of standard characterization performance data. The philosophy is that these parameters have a physical background, so that once correlated, the model can be used to predict compressor performance under operating conditions which are not tested, for example, at extreme temperatures or at lower speeds. Furthermore, the correlated model may estimate the compressor performance with other refrigerants for which there are no available data. In the literature, one can find other models which follow this same approach (see Refs. [12,14]). The present study attempts to move forward towards that objective, targeting the selection of the parameters in such a way that they retain the maximum physical significance. The values obtained in the correlations are expected to show a clear agreement with the reasonable orders of magnitude of the compressor characteristics they represent. Additionally, the fitting techniques employed have been shown to have a considerable influence on the suitability of the obtained coefficients. The authors found that classic least square correlation methods are not useful in this kind of non-linear system. Monte Carlo based fitting methods, on the contrary, provide much greater freedom and stability in the definition of the model parameters, better final results and a better way of avoiding excess sensitivity problems. This first paper is structured in three main parts. First, the model is presented; then the Monte Carlo fitting procedure is described and finally the model is validated for a compressor providing an initial assessment of the order of magnitude of the obtained parameters. 2. Compressor model for a reciprocating compressor Examining the evolution of the refrigerant in a peh diagram, the model will assume the evolution shown in Fig. 1. Fig. 1. Refrigerant cycle inside the compressor. The refrigerant enters the compressor at point 1 (‘‘inlet conditions’’). The reference for the compressor efficiency is given by an isentropic condition from the inlet to the outlet of the compressor: 1e8*. The real conditions at the outlet of the compressor are indicated by state 8 in Fig. 1. The developed model assumes that the evolution of the refrigerant through the compressor can be divided into the following sequence of effects:  1e2: Vapor heating due to motor cooling and mechanical loss dissipation.  2e3: Vapor heating due to the heat transferred from the hot side of the compressor (discharge) to the inlet flow and to the leaks.  3e4: Isoenthalpic pressure lost at the suction valve.  4e5: Isentropic compression from real cylinder intake conditions (leaks and possible condensation also appear in this part of the process).  5e6: Isoenthalpic pressure lost at the discharge valve.  6e7: Vapor cooling due to the heat transferred to the suction side. Regarding the evolution from 4 to 5, inside the cylinder, measurements from Ref. [16] show that the real compression from conditions at the bottom of the piston (beginning of the compression stroke) to the top dead centre is very close to isentropic. The main source of irreversibilities is the heat transfer to and from the wall during the compression stroke. However, the process is very fast and wall temperatures are quite close to fluid temperatures; thus, heat transfer effects per mass flow rate unit are slight. For this reason, a generalized polytropic compression was avoided since the polytropic exponent must be fitted for each refrigerant. The internal leakage of refrigerant through the piston rings has a considerable effect on compressor and volumetric efficiencies. In order to simplify the treatment of this loss, an evaluation of the leak is made at state 5. It is considered as if the loss of the circulating mass flow rate ðm_ leak Þ takes E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 place at state 5. Therefore, the compressor is assumed to consume the work of compression for the circulating mass flow rate plus leaks ðm_ in þ m_ leak Þ. The possibility of condensation of a relatively significant fraction of refrigerant during compression was also evaluated throughout this study. The condensed mass does not flow towards the discharge, but it is later evaporated during the suction stroke. The effect of this possible condensation is similar to an increase of the dead space ratio. In the model, this possible effect is treated as a loss of the circulating mass flow rate ðm_ pc Þ. Other effects that influence refrigerant temperature before the refrigerant leaves the compressor (7e8 in Fig. 1) such as heat release to the environment or oil heating and electric motor heating of the vapor are not considered in this study. With these assumptions, a model describing compressor and volumetric efficiencies of the compressor is proposed. The circulating mass flow rate can be calculated from the ideal flow rate by means of the expression: m_ in ¼ hsth V_ s r4  m_ leak  m_ pc ð1Þ In expression (1), V_ s is the swept volume flow given by _V s ¼ n$Vs , Vs being the swept volume and n the nominal speed of the compressor, and hsth ¼ 1  ðVd =Vs Þ ððr8 =r1 Þ  1Þ is the ideal volumetric efficiency [17], where it is supposed that the density ratio between states 1 and 8 is not very different from the density ratio between states 4 and 5 which is closer to the real one. It should be noted here that the compressor nominal speed was considered constant throughout the study. For a line frequency of 50 s1, the manufacturer estimates a value of 2900 rpm for these compressors. Expression (1) implies that the total mass flow rate is given by the total cylinder refrigerant capacity under conditions corresponding to point 4 in Fig. 1, substracting: the cylinder volume losses from the cylinder clearance, the vapor that is leaked during the compression process and the possible formation of small refrigerant droplets in some part of the cylinder surface that are not subsequently pumped out of the cylinder during the compression process. Electric compressor power input E_ k can be assumed to be the energy that the refrigerant consumes to change from state 4 to state 5 in Fig. 1 plus the energy that the compressor con_ mech Þ and electrical losses. So E_ k sumes in mechanical ðEL can be expressed as: i   1h _ mech E_ k ¼ Dhð4e5Þ m_ in þ m_ leak þ EL hel ð2Þ In expression (2) the term related with a possible phase change phenomenon does not appear because, as explained later, it is assumed that the energy given by the system to the refrigerant to allow the condensation comes back to the system in the reexpansion part of the process. 1257 From Eqs. (1) and (2), the corresponding expressions for the volumetric and compressor efficiencies are obtained: hs ¼ m_ in r m_ leak m_ pc ¼ 4h   _V s r1 r1 sth V_ s r1 V_ s r1 ð3Þ hk ¼ m_ in Dhð1e8 Þ ¼ E_ k ð4Þ Dhð4e5Þ Dhð1e8 Þ hel ! _ mech m_ leak EL þ 1þ m_ in m_ in The different processes considered in expressions (3) and (4) will now be described in further detail. 2.1. Vapor heating due to motor cooling and mechanical loss dissipation The heating of the inlet refrigerant by motor cooling and mechanical loss dissipation is quantified by the following expression: _ mech Zmvapor Q_ 1e2 ¼ ð1  hel ÞE_ k Zelvapor þ EL where Zelvapor and Zmvapor are the fractions of the losses transferred to the suction vapor as heat. It is assumed that the fraction of absorbed heat is the same for both losses, that is Zelvapor ¼ Zmvapor, so these factors can be renamed as K1. The remaining heat is released to the environment either through the outlet gases or through the compressor surface. The increase in the suction vapor temperature 1e2 is thus given by: DT1e2 ¼   _ mech ð1  hel Þ Dhð1e8 Þ EL Q_ 1e2 þ ¼ K1 hk cp1 m_ in cp1 V_ s hs r1 cp1 ð5Þ 2.2. Vapor heating due to heat transferred from the hot side of the compressor (discharge) to the inlet flow Before leaving the compressor, the hot vapor flowing outside the cylinder heats the refrigerant at the low pressure side. As a first approximation, the heat transferred between both sides can be given by Q_ 2e3 ¼ ðUAÞht ðT8  T1 Þ, where the temperature difference ðT8  T1 Þ, calculated from the isentropic e ideal process, is considered as an effective temperature difference, characteristic of the process. The global heat transfer coefficient Uht is related to the heat transfer coefficient hht using these approximations:  The heat transfer between both sides takes place mainly in the suction and discharge pipes near the cylinder.  Uht is considered proportional to the heat transfer coefficient hht, Uht ¼ C0 $ hht.  The heat transfer coefficient hht is considered as the one given for the turbulent internal flow in a pipe. 1258 E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 Nu ¼ C$Re0:8 Pr0:4 /hht ¼ 0:8   k m$cp 0:4 V_ s $hs $r1 $C$ dh dh $m k ð6Þ Finally, considering these approximations, the temperature increase is expressed as follows: DT2e3 ¼ K2 ðT8  T1 Þ k20:6 ðT8  T1 Þ k20:6 ¼ K20 0:2 1:8 0:6 0:4 0:4 ðV_ s hs r1 Þ dh cp2 m2 ðV_ s hs r1 Þ0:2 c0:6 p2 m2 ð7Þ where dh is a hydraulic diameter characteristic of the narrow flow passages around the cylinder. If no reasonable estimation of these passages is available, the hydraulic diameter can be included within a new constant K20 ¼ K2 =dh1:8 . e 5 D2 Þ, as done for proportional to the cylinder area ðAL zK the compressor inlet and outlet valve areas. The pressure difference DPm ¼ K52 ðP8  P1 Þ is assumed to be proportional to the overall pressure difference ðP8  P1 Þ and the mean pffiffiffiffiffiffiffiffiffiffiffi density for leaked vapor rm z r8 r1. With these assumptions m_ leak is given by: pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi m_ leak ¼ K50 $nz$ DPm rm ð11Þ e 5 $z1=2 : where K50 ¼ K5 $D2 $K _mleak influences the reduction of the total mass flow rate (see Eq. (1)) and also affects the refrigerant temperature in the cylinder inlet, which is estimated as the corresponding mixing temperature: m_ leak ðT8  T2 Þ m_ in þ m_ leak 2.3. Isoenthalpic pressure losses at the inlet valve DT2e20 ¼ This pressure drop is estimated as DP3e4 ¼ x3 r3 ðw23 =2Þ. Expressing the velocity as w3 ¼ V_ s hs =nzAc3 where nz is the number of cylinders and Ac3 the effective inlet valve area per cylinder, the pressure drop is given by:  2 V_ s hs DP3e4 ¼ K3 r3 ð8Þ nz The use of more accurate expressions, like the compressible flow equation for unchoked flow e maybe more precise e would not significantly improve the results of the model, as commented in Ref. [18], and they would certainly lead to much longer computation times. The authors consider that the above approximation is a reasonable compromise between simplicity and physically relevant information regarding the internal process in the present model. where K3 is given by K3 ¼ x3 =2A2c3. From a practical point of view, it may be useful to express the inlet valve area Ac3 in terms of the cylinder diameter e 3 $D2 . Thus, if the value of K3 is known for one comAc3 ¼ K pressor, an estimation can be made for other compressors of similar characteristics but different cylinder diameters. 2.4. Isoenthalpic pressure losses at the outlet valve Using the same arguments as in the previous section, the pressure loss for the outlet valve is given as:  2 r V_ s hs DP5e6 ¼ K4 r5 4 ð9Þ r5 nz where K4 is given by K4 ¼ x5 =2A2c5, and Ac5 is the effective outlet valve area per cylinder. 2.5. Leakages To describe leakages, it is assumed that they are a function of the overall pressure difference in the compressor and are not dependent on the vapor flow. The mass flow rate leaked is deduced considering that the refrigerant is a nonpffiffiffiffiffiffiffiffiffiffiffi compressible fluid with a density rm ¼ r8 r1, resulting in: sffiffiffiffiffiffiffiffiffi DPm r ð10Þ m_ leak ¼ nz$AL K5 zrm m where z represents the equivalent flow resistance, and AL is the effective leakage area per cylinder. AL can be considered ð12Þ 2.6. Refrigerant phase changes inside the cylinder Some refrigerant liquid droplets can be formed on the colder areas of the cylinder. These droplets can be in continuous phase change, leading to a reduction in the total amount of vapor flowing through the compressor and thus affecting volumetric efficiency. The inlet valve is exposed to relatively cold vapor on the suction side; therefore, its temperature could fall below the dew point of the refrigerant during some fraction of time of the compression stroke. Thus, condensation may occur and some droplets could appear on the valve plate. These droplets would then evaporate during suction and, assuming that this evaporation takes place on the valve plate, a cooling of the valve plate would take place to allow again condensation in the subsequent cycle. It is worth noting that the condensation part of this phase change process is quite a critical point because it has very little time to be produced (the pressure inside the cylinder is only higher than the refrigerant dew point for some fractions of a second). Considering that the most important factor to facilitate this process is the temperature difference between the cylinder inlet and outlet, the temperature difference ðT8  T1 Þ may be considered as an effective temperature difference to characterize this process. The amount of condensing refrigerant for one compression cycle of the cylinder may be expressed as: m_ pc ¼ t$nz$hpc $Av ðT8  T1 Þ hfg ðT1 Þ ð13Þ E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 where t is the fraction of time during the compression in which the condensation can be produced (fraction of time in which the pressure in the cylinder is higher than the dew point for the inlet temperature) and hpc is an effective heat transfer coefficient, considered as approximately constant. Expression (14) to be inserted into the model must be expressed in terms of the mass flow per time unit and not per cycle: m_ pc ¼ t$nz$hpc $Av ðT8  T1 Þ nzðT8  T1 Þ ¼ K6 hfg ðT1 Þ hfg ðT1 Þ ð14Þ where t represents the percentage of the total cycle in which the pressure is high enough to produce the refrigerant condensation. Regarding this phase change effect, previous researches can be found in literature, for instance, in Ref. [19] for low speed piston compressors working in wet conditions. Nevertheless, there is no experimental information available regarding this phenomenon in the present research and here, it is more intended as a proposal to explain some model results and to inspire further research. 2.7. Mechanical loss influence on the compressor efficiency According to Ref. [20], the mechanical losses may be considered as a sum of two terms, one proportional to the energy consumption rate and the other dependent on the compressor speed: _  _ mech ¼ K7 E_ k þ K8 n2 ¼ K7 V s hs r1 Dhð1e8 Þ þ K8 $n2 EL hk ð15Þ 2.8. Final model structure To formulate the global model, the equations governing the different losses described in Sections 2.1e2.6 are either introduced into Eqs. (3) and (4) by direct substitution of the obtained equations (Eqs. (11), (14) and (15)) or are used to calculate the refrigerant states 4 and 5 (Eqs. (5), (7)e(9) and (12)). This leads to a system of two implicit equations for the compressor and volumetric efficiencies:   V0 f1 hk ; hs ; G; K; hel ; ; P1 ; P8 ; DTsh ¼ 0 ð16Þ Vs   V0 f2 hk ; hs ; G; K; hel ; ; P1 ; P8 ; DTsh ¼ 0 ð17Þ Vs where K (K ¼ (K1, ., K8)), hel ; and V0 =Vs represent compressor design parameters, which are difficult to determine, whereas G (G ¼ (G1, .Gn)) stands for compressor design parameters that are easy to know like stroke (S), number of cylinders (nz), nominal speed (n), and the like. If for any reason some of these G parameters were not known, they can be easily regrouped inside the K parameters. 1259 Once all the compressor design parameters K are known, the system of two equations can be solved for compressor and volumetric efficiencies, hk and hs, for a given working condition ðP1 ; P8 ; DTsh Þ. Any solver for a system of nonlinear equations can be employed. The results shown in this paper were obtained by the GausseSeidel procedure [21]. With this algorithm, given an initial value of 0.5 for compressor and volumetric efficiencies, the convergence to the solution is typically reached in less than 15 iterations. As commented above, the parameters K; hel ; V0 =Vs are difficult to estimate. A set of data for a number of working conditions obtained either from experiments or from manufacturer catalogs is required to obtain the proper value of K by a fitting procedure. The developed fitting procedure to find the best estimation of K is explained in Section 3. From the best obtained values of K, it is then possible to determine the value of compressor and volumetric efficiencies in conditions different from those tested. Besides, it is possible to obtain physically relevant information about the internal processes inside the compressor and to quantify the different losses. A key assumption regarding K is that the different parameters are not significantly dependent on the test conditions or employed refrigerants. The obtained results indicate that this assumption is fairly reasonable. 3. Statistical fitting procedure As explained in Section 2, the last step to close the model is to estimate the compressor losses parameters K from experimental or catalog data. This is quite a critical issue in these kinds of models because the dependency of the target functions on the parameters is non-linear, and there is also some sort of indetermination, in the sense that several possible combinations of parameters could adjust the model properly with a final deviation in the predicted efficiency values which is smaller than the experimental uncertainties. In fact, several conventional non-linear regression techniques (the standard routines in MsExcel, Origin, simplex algorithm [21]) were tested, yet they failed in the fitting process of the proposed model. For this reason and to avoid possible problems arising from a step-by-step exploration of the parameter space (existence of a local minimum solution, too smooth dependence on certain parameters and the like), a heuristic algorithm based on a Monte Carlo type approach was designed. A review of the main trends in this field can be found in Ref. [22]. Although computationally not the most efficient these methods show great versatility and reliability. A general scheme of the designed algorithm is shown in Fig. 2. In this scheme, the program starts by assigning pseudo-random values (according to the uniform distribution) to the parameters and the ‘‘best’’ combination of them to fit the compressor and volumetric efficiencies data is sorted out (this is called the first process). The routine used to generate the pseudo-random numbers was the one 1260 E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 Fig. 2. Fitting procedure algorithm. proposed by Park and Miller [23]. This routine has a long enough period for this specific application. To select the ‘‘best’’ combination of parameters, an error function or residue (e) must be defined. In this case, the Euclidean norm weighted by the standard deviation of each experimental point i was selected: e¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi X h2k ðxi Þ  h2kexp ðxi Þ i sðxi Þ þ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi h2s ðxi Þ  h2sexp ðxi Þ sðxi Þ The set of parameters with a lower value of e is selected as a solution to the first process. Nevertheless, as a consequence of experimental errors, intrinsical errors associated to the model and the nature of the mathematical functions involved, if the process is repeated, a different set of parameters is obtained as a solution which may also give a good value for the error function e. Therefore, this process is repeated until a representative map of the solutions in the parameter space is obtained (this is called the second process). This means that the probability distribution of the solutions for each parameter is obtained. The result of this process is a set of probability distributions for each of the model parameters. From these distributions, the most probable value for each parameter is selected as the best fit value. Some comments should be made regarding this scheme:  An interval in which the value of the parameters must be found, must be defined. As a result of the intrinsical stability of the method, this is not a critical point in the model, yet a good selection of this interval reduces the number of iterations needed to find a suitable solution.  Preliminary studies have been developed to determine the proper number of iterations in the first and second processes. For the first process, this number is reached only if, on increasing the number of iterations, the order of magnitude of e does not change. For the second process, this number is reached if, on increasing the number of iterations, the obtained parameter distribution function does not change.  To reduce the high computational cost linked to the direct use of REFPROP [24] in the evaluation of the thermodynamical properties of the refrigerants, an approximation based on linear interpolations of 1261 E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 bidimensional meshes from REFPROP was employed (see Ref. [25] for details). 4. Results from the analysis of a two-piston hermetic compressor To validate the model and the fitting methodology, a twocylinder reciprocating compressor working with propane was analyzed using the set of experimental data from Ref. [26] (compressor ST, in the aforementioned nomenclature); all the information related to the accuracy and the way in which these data were obtained can be found in that reference. A comparison between the obtained results with and without the phase change term is also presented to understand the possible relevance of this term. After an initial study of possible correlations among the different parameters K for several compressors, it was found that K1 and K2 showed some kind of coupling, which can only be solved with a very high number of trial points (in the first process of the Monte Carlo method). In order to keep the fitting time moderate and after finding that the value of K1 was almost the same for several compressors, the criterion of assigning a constant value of 0.9 provided very good results. Electric efficiency is a function of the operating conditions, yet this dependence is usually small in the nominal operation range of the compressor. For this study, information from the manufacturer about experimental electric motor efficiency was available. This information shows a maximum deviation of 1% over the nominal value of the electric efficiency in almost all the tested points, so that electric efficiency can be considered independent of the operation conditions throughout the whole study. Although this parameter was known, to check the model capacities, this parameter was left free during the fitting process because it is usually not a parameter available in catalogs. Table 1 shows the value obtained for parameters K in both cases (including or not including the phase change term), together with the interval in which the parameters were searched. The search interval was chosen to include all the possible values that the compressor parameters might have in an attempt to cover the widest possible range bounded only by the physical constraints. Regarding the Monte Carlo methodology, 1,000,000 trial runs were required to obtain a stable value for the error function in the first process and 1000 iterations were needed in the second process to obtain a representative distribution of probability of the space of parameters. The values obtained for the parameters, despite the large number of assumptions involved in the model, provide a very good prediction of the compressor performance throughout the entire test sample. Further, they have values consistent with the compressor geometry and the physical process involved, as described in the following:  K2: As seen in Table 1, the obtained value for K20 is 2.80, if a value of 0.023 for the constant C in expression (6) is taken (DittuseBoelter correlation [27]), and considering that Uwh=2, the constant K20 represents the relation between the effective heat transfer area and the effective hydraulic diameter and is given by the term K20 ¼ ð0:023$40:8 =2$ðpÞ0:8 Þ ðAht =dh1:8 Þ. Therefore, the relation between both magnitudes is Aht =dh1:8 w215. The values of dh and Aht are difficult to estimate, but in any case, it is reasonable to think that dh should be quite small. If this is so, the value obtained for K20 supports the hypothesis that the heat transfer area of this process should be small.  K3: The measured value for the inlet valve orifice section Ac4 was 160 mm2, assuming that it is approximately the value of the valve flow section. The value obtained for K3 gives a value for the drag factor of the inlet valve Table 1 Definition and obtained value of each model parameter for the compressor studied K1 K20 ðm5 Þ K3 (m2) K4 (m2) K50 ðm2 Þ Vd/Vs K6 K7 K8 (kW) hel Parameter definition Model without phase change Model with phase change inside the cylinder Search interval K1 Aht C0 $C$ 1:8 dh x3 2A2c3 x4 2A2c4 AL $K5 2 pffiffiffi $D z Vd/Vs t$hpc $Av K7 K8 hel 0.90 2.80 0.90 2.69 [d] [0e18] 1.94  107 1.89  107 [0e8.74  107] 3.85  108 3.4  108 [0e2.5  109] 0.95  106 0.85  106 [0e4  105] 6.77  102 0.0 5.11  102 0.2052 0.859 3.90  102 1.73 4.64  102 0.2051 0.862 [0e0.4] [0e18] [0e1] [0e3.73] [0.5e1]    (1) (2) E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 of x ¼ 0.99, which is quite reasonable considering the approximations involved in the process. K4: The measured value for valve orifice section Ac5 was 100 mm2, assuming that it is approximately the value of the valve flow section. The value obtained for K4 also gives a value of x ¼ 3.77 for the drag factor of the outlet valve. This value seems too high, probably it is associated with the fact that the real effective area of the outlet valve is smaller than the geometrical one, considered in this estimation. K5: In Ref. [16], an estimation of the leaks was made using the Saint-Venant equation for compressible flow, and the value obtained for the effective leaks area was AL w 1 $ 106 m2. This value is consistent with the one obtained in the developed model for the constant pffiffiffi K5 AL = z. K6: In Ref. [26] a value of 0.037 was given for the dead space ratio of the compressor, the value obtained for the dead space ratio in the model with no phase change inside the cylinder was 0.068. This value was considerably larger than that expected (0.037); thus, the possible condensation of some fraction of refrigerant over a possible cold spot in the cylinder head was considered to explain this loss of mass flow. This assumption resulted in an estimation of 0.039 for the dead space ratio. This value was quite near the expected value. In the model with a phase change inside the cylinder, the value obtained for the constant representative of this process was K6 ¼ t$hpc $Av ¼ 1:73. Considering, as a rough estimation, that: The moment in which the condensation can occur is tw0:12 ¼ 12% of the total time of one revolution. This time is supposedly equal to the time of the process of delivering vapor through the outlet valve (period of time when the pressure inside the piston is higher). The area Av in which this process can be produced is the inner surface of the inlet valve (Av w 7 $ 104 m2). Fig. 3. Calculated and experimental compressor efficiencies. Experimental compressor efficiency 1262 0.70 0.65 5% Model Model with phase change 0.60 -5% 3% 0.55 -3% 0.50 0.45 0.45 0.50 0.55 0.60 0.65 0.70 Calculated compressor efficiency Fig. 4. Comparison between calculated and experimental compressor efficiencies. The obtained heat transfer coefficient is hpc ¼ ðK6 =Av tÞ w20; 000 W m2 K1 ; it is quite reasonable if, for example, a dropwise condensation phenomenon is produced, since Refs. [28] and [29] reported heat transfer coefficients for dropwise condensation up to 300,000 W m2 K1 and Brown [19] reported values of the heat transfer coefficient for freon-12 in dropwise condensation of the same order of magnitude as that obtained in this work.  K7, K8: The obtained values for the mechanical losses, approximately 10%, are in agreement with the expected value for this type of compressor.  hel: This is the most influential factor in compressor efficiency. The obtained value for hel is 0.859, which is very close to the actual mean electric motor efficiency data (hel ¼ 0.862). Compressor and volumetric efficiencies obtained using both model versions and their relative errors are shown in Fig. 5. Calculated and experimental volumetric efficiencies. E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 1263 Fig. 6. Comparison between calculated and experimental volumetric efficiencies. Fig. 7. Comparison between calculated and measured compressor outlet temperatures. Figs. 3e6. The differences between model results, using the best fit parameter values, and the experimental data are always lower than 5%. The Pearson correlation coefficients obtained for the line that represents no divergence between experimental and calculated compressor and volumetric efficiencies in Figs. 3e6 are shown in Table 2. The results show that both versions of the model (with and without phase change) reproduce the experimental results accurately enough. The values obtained for the different parameters in both models are in the same range. In brief, the effect of the phase change parameter seems to be equivalent to an increase in the dead space. In Fig. 7, the vapor temperatures at point 7 of the pressuree enthalpy diagram (Fig. 1) are plotted against measured compressor outlet temperatures. The temperature given by the model in state 7 of this figure, in principle, can only be considered as a rough estimation of the real temperature at the exit of the compressor (point 8 in Fig. 1). In any event, this estimation is actually quite good for low and medium pressure ratios where the temperature of the refrigerant is not high, and it is worse at high pressure ratios, where the temperature of the refrigerant is higher and some processes not considered in the model, like the heat transfer of the hot refrigerant to the crankcase, by radiation, could become more relevant. 5. Conclusions Table 2 Correlation coefficients obtained for the compressor and volumetric efficiencies for both versions of the model hk hs Model without phase change Model with phase change inside the cylinder 0.932 0.996 0.943 0.994 A model for reciprocating compressors was developed. This model can reproduce the compressor and volumetric efficiencies with an error lower than 3% under a wide range of operating conditions. Considering the main sources of losses, the model is based on an ideal evolution of the refrigerant throughout the compressor. This model has 10 empirical parameters, each with a direct physical interpretation. If these parameters are unknown, they must be fitted with some empirical data. To this end, a statistical fitting methodology based on Monte Carlo techniques was designed. This methodology was tested on one compressor and the results with 16 experimental points are quite good. To apply the developed fitting methodology, only data commonly available in catalogs are required. One drawback of the developed model is that the volumetric and compressor efficiencies are presented as a system of implicit equations. Fortunately, this is not a major drawback nowadays, given the existence of many numerical solvers adequate for this kind of system. Although the developed fitting methodology is very stable, the computational time involved in the fitting process is quite long (6 h in a Pentium IV processor, 3.4 GHz, 1 GB RAM). All model parameters have a direct physical interpretation and characterize the design and performance of the compressor. In general, the developed model could be quite useful in: (1) Estimating compressor performance at operating points, different from the experimental points used on the fit. Furthermore, the model could be used to estimate the compressor performance with another refrigerant, with other compressor speeds or with slight modifications in the cylinder geometry, for instance. 1264 E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 (2) Characterizing the compressor performance with 10 parameters and analyzing their adequacy from their absolute value or by comparison with a set of reference parameters. For example, unusual values of motor electric efficiency, mechanical losses or valve losses, can point to internal compressor malfunctions. The value for the dead space ratio found for the studied compressor is too large (6.9%). Several effects were evaluated to explain this result (valve dynamic effect, increase of the leaks) but the kinds of terms tested interact simultaneously with compressor and volumetric efficiencies reducing the correlation factors between the experimental and modelled efficiencies. Considering the possibility of a condensation phenomenon on a cold spot inside the cylinder, like the inner surface of the inlet valve, during the end of the compression stroke leads to a value of this parameter (3.9%) more similar to the real value (3.7%). A study of the order of magnitude of this process revealed that a dropwise condensation could explain this process. Together with similar results obtained from other compressors studied, this result seems to support the possible existence of such a phenomenon. In Ref. [15], the model proposed here will be used to analyze a set of hermetic reciprocating compressors working with propane and the capabilities of the model to predict compressor behavior using other refrigerants will be evaluated. [7] [8] [9] [10] [11] [12] [13] [14] [15] Acknowledgements This work has been partially funded by the Spanish Ministerio de Educacion y Ciencia through project, ref. ENE 2004-04551 (GEOCARE). The authors also express their gratitude to Debra Westall for her kind cooperation. References [1] ARI Standard 540, Standard for positive displacement refrigerant compressors and compressor units, 1999. [2] Proceedings of the International Compressor Engineering Conferences at Purdue, 1996, 1998, 2000, 2002. [3] F.F.S. Matos, A.T. Prata , C.J. Deschamps, A numerical methodology for the analysis of valve dynamics, Proceedings of the International Compressor Engineering Conference, 2000, Purdue, USA, pp. 391e396. [4] M.D. Libera, A. Faraon, A. Solari, A complete analysis of dynamic behaviour of hermetic compressor cavity to improve the muffler design, Proceedings of the International Compressor Engineering Conference, 2000, Purdue, USA, pp. 665e669. [5] Y.C. Ma, O.K. Min, On study of pressure pulsation using a modified Helmholtz method, Proceedings of the International Compressor Engineering Conference, 2000, Purdue, USA, pp. 657e664. [6] F. Fagotti, A new correlation for instantaneous heat transfer between gas and cylinder in reciprocating compressors, [16] [17] [18] [19] [20] [21] [22] [23] [24] Proceedings of the International Compressor Engineering Conference, 1998, Purdue, USA, pp. 871e876. Copland catalogue select 5.<www.ecopland.com>. W. Soedel, Introduction to computer simulation displacement type compressors, Purdue University Short Courses, 1972, IN, USA. R. Prakash, R. Singh, Mathematical modelling and simulation of refrigerating compressors, Proceedings of the International Compressor Engineering Conference, 1974, Purdue, USA, pp. 274e285. J.M. Corberan, J. Gonzalvez, J.F. Urchueguı́a, A. Calas, Modelling of refrigeration piston compressors, Proceedings of the International Compressor Engineering Conference, 2000, Purdue, USA, pp. 571e578. C.D. Pérez-Segarra, J. Rigola, A. Oliva, Modelling and numerical simulation of the thermal and fluid dynamic behaviour of hermetic reciprocating compressors. Part 1: theoretical basis, International Journal of Heating Ventilation, Air Conditioning, and Refrigeration Research 9 (2003) 215e236. A. Mackensen, S.A. Klein, D.T. Reindl, Characterization of refrigeration system compressor performance, Proceedings of the International Refrigeration Engineering Conference, 2002, Purdue, USA, pp. R9-1. P. Popovic, H.N. Shapiro, A semiempirical method for modelling a reciprocating compressor in refrigeration systems, ASHRAE Transactions 101 (2) (1995) 367e382. E. Wynandi, O. Saavedra, J. Lebrun, Simplified modelling of an open-type reciprocating compressor, International Journal of Thermal Sciences 41 (2002) 183e192. E. Navarro, E. Granryd, J.F. Urchueguı́a, J.M. Corberan, Performance analysis of a series of hermetic reciprocating compressors working with R290 (propane) and R407C, International Journal of Refrigeration 30 (7) (2007) 1254e1265. J. Gonzalvez, Desarrollo de un modelo global de compresores de refrigeración de desplazamiento positivo, Doctoral thesis, Universidad Politécnica de Valencia, 2001. E. Granryd, I. Ekroth, P. Lundqvist, A. Melinder, B. Palm, P. Rohlin, Refrigerating Engineering, KTH, Stockholm, 1999. J. Prins, Selected basic theory of gas leakage, Proceedings of the International Conference on Compressors and Their Systems, 2003, London, England. J. Brown, Suction conditions: its effect on refrigeration compressor performance, Doctoral thesis, Faculty of Engineering, University of Glasgow, 1963. J.P. Bourdhouxhe, M. Grodent, J. Lebrun, C. Saavedra, K. Silva, A toolkit for primary HVAC system energy calculation e part 2: reciprocating chiller models, ASHRAE Transactions 100 (2) (1994) 774e786. W. Press, S.A. Teukolsky, Numerical Recipes, American Institute of Physics Inc., 1989. P.C. Sabatier, Past and future of inverse problem, Journal of Mathematical Physics 41 (2000) 4082e4124. S.K. Park, K.W. Miller, Random number generators: good ones are hard to find, Communications of the ACM 31 (10) (1988) 1192e1201. M.O. McLinden, S.A. Klein, E.W. Lemmon, A.P. Peskin, NIST reference fluid thermodynamics and transport properties ‘‘REFPROP’’, National Institute of Standards and Technology, Gaithersburg, U.S. Department of Commerce, 2002. E. Navarro et al. / International Journal of Refrigeration 30 (2007) 1254e1265 [25] J. Corberan, J. Gonzalvez, D. Fuentes, Calculation of refrigerant properties by linear interpolation of bidimensional meshes, IIR International Conference of Thermophysical Properties and Transport Process of Refrigerants, 2005, August 31eSeptember 2, Vicenza, Italy. [26] E. Navarro, J.F. Urchueguı́a, J. Gonzalvez, J.M. Corberan, Test results of performance and oil circulation rate of commercial reciprocating compressors of different capacities 1265 working with propane (R290) as refrigerant, International Journal of Refrigeration 28 (2005) 881e888. [27] T.W. Dittus, L.M.K. Boelter, Public Engineering 2 University of California, Berkeley, 1930, pp. 443. [28] G.F. Hewitt, G.L. Squires, Y.V. Polzhaev, International Encyclopedia of Heat and Mass Transfer, CRC Press LLC, 1997. [29] A.F. Mills, Heat and Mass Transfer, Richard D. Irwin, 1994, ISBN 0256114439.